Numerical Solution of System of Fredholm-Volterra Integro-Differential Equations Using Legendre Polynomials

In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. The equation considered in this paper involves the derivative of unknown functions in the integral term, which makes its numerical solution more complicated. We first introduce a single-step Legendre collocation method on the interval [0, 1]. Next, a multi-step version of the proposed method is derived on the arbitrary interval [0, T] that is based on the domain decomposition strategy and specially suited for large domain calculations. The first scheme converts the problem to a system of algebraic equations whereas the later solves the problem step by step in subintervals and produces a sequence of systems of algebraic equations. Some error estimates for the proposed methods are investigated. Numerical examples are given and comparisons with other methods available in the literature are done to demonstrate the high accuracy and efficiency of the proposed methods.